The operator \(L_j\)  can be seen as a natural substitute for \(D^j\) ; see [1]}. In particular, we have
\(L_j g_k(x) = {\left\lbrace \begin{array}{ll}w_{j}(x) \int _{z}^x w_{j+1}(y_{j+1}) \int _{z}^{y_{j+1}} \cdots \int _{z}^{y_{k-1}} w_k(y_k) {\rm d}y_{k} {\rm d}y_{k-1}\cdots {\rm d}y_{j+1}, & k=j,\ldots ,p,\\0, & k=0,\ldots ,j-1.\end{array}\right.}\) 
